Quadrifolium



The quadrifolium (also known as four-leaved clover ) is a type of rose curve with an angular frequency of 2. It has the polar equation:


 * $$r = a\cos(2\theta), \,$$

with corresponding algebraic equation


 * $$(x^2+y^2)^3 = a^2(x^2-y^2)^2. \,$$

Rotated counter-clockwise by 45°, this becomes


 * $$r = a\sin(2\theta) \,$$

with corresponding algebraic equation


 * $$(x^2+y^2)^3 = 4a^2x^2y^2. \,$$

In either form, it is a plane algebraic curve of genus zero.

The dual curve to the quadrifolium is


 * $$(x^2-y^2)^4 + 837(x^2+y^2)^2 + 108x^2y^2 = 16(x^2+7y^2)(y^2+7x^2)(x^2+y^2)+729(x^2+y^2). \,$$



The area inside the quadrifolium is $$\tfrac 12 \pi a^2$$, which is exactly half of the area of the circumcircle of the quadrifolium. The perimeter of the quadrifolium is
 * $$8a\operatorname{E}\left(\frac{\sqrt{3}}{2}\right)=4\pi a\left(\frac{(52\sqrt{3}-90)\operatorname{M}'(1,7-4\sqrt{3})}{\operatorname{M}^2(1,7-4\sqrt{3})}+\frac{7-4\sqrt{3}}{\operatorname{M}(1,7-4\sqrt{3})}\right)$$

where $$\operatorname{E}(k)$$ is the complete elliptic integral of the second kind with modulus $$k$$, $$\operatorname{M}$$ is the arithmetic–geometric mean and $$'$$ denotes the derivative with respect to the second variable.